Given any background (or seed) solution of the nonlinear Schrodinger equation, the Darboux transformation can be used to generate higher-order breathers with much greater peak intensities. The Darboux transformation is generic in iterating a pair of generating solutions of the Lax- pair equation which preserves the solution of the nonlinear Schrodinger equation as a consistency condition. The Darboux transformation itself knows nothing about the background solution except through the initial pair of Lax solutions. Because of this, in this work, we can prove in an unified manner, and without knowing the analytical form of the background solution, that the peak-height of a high-order breather, is just a sum of peak-heights of first-order breathers plus that of the background, irrespective of the specific choice of the background. Detailed results are verified for breathers on a cnoidal background. Generalizations to more extended nonlinear Schrodinger equations are also indicated.